Propagation of chaos for mean field rough differential equations

TL;DR

This paper proves propagation of chaos for large systems of mean field rough differential equations driven by random rough paths, providing explicit convergence rates using coupling methods.

Contribution

It introduces a novel approach to propagation of chaos for mean field rough differential equations with explicit convergence rates.

Findings

Established propagation of chaos for mean field rough differential equations.

Derived explicit optimal convergence rates.

Utilized coupling arguments adapted from particle systems with Brownian inputs.

Abstract

We address propagation of chaos for large systems of rough differential equations associated with random rough differential equations of mean field type $$ dX_t = V(X_t,\mathcal{L}(X_t))dt + F(X_t,\mathcal{L}(X_t))dW_t $$ where $W$ is a random rough path and $\mathcal{L}(X_t)$ is the law of $X_t$. We prove propagation of chaos, and provide also an explicit optimal convergence rate. The analysis is based upon the tools we developed in our companion paper [1] for solving mean field rough differential equations and in particular upon a corresponding version of the It\^o-Lyons continuity theorem. The rate of convergence is obtained by a coupling argument developed first by Sznitman for particle systems with Brownian inputs.